243
The (digital) life of Pi
Jonathan M. Borwein and Mason S. Macklem
From 23 September till 20 October 2005, The Australian Mathematical Society and the
International Centre of Excellence for Education in Mathematics (ICE–EM) invited the
Canadian mathematician Jonathan M. Borwein http: // users. cs. dal. ca/ ~jborwein/
for providing expert advice on the use of Access Grid Rooms to overcome Australia’s ‘tyranny
of distance’ in education, business and industry.
Jonathan has become widely known as an advocate for experimental mathematics, a field
in which the extensive computational power of modern day computers is used to discover
mathematical theorems. The article below, joint with Mason S. Macklem, discusses many of
the digital truths about π, together with the computational and algorithmic advances that
have been made discovering these truths.
The desire to understand π, the challenge, and originally the need, to calculate ever more
accurate values of π, the ratio of the circumference of a circle to its diameter, has challenged
mathematicians–great and less great—for many many centuries. Recently, π has provided
compelling examples of computational mathematics. It is also part of both mathematical
culture and of the popular imagination.
Why computations of π? One historical motivation was very much in the spirit of modern
experimental mathematics: to see if the decimal expansion of π repeats, which would mean
that π is the ratio of two integers (i.e., rational), or to recognize π as algebraic—the root of a
polynomial with integer coefficients—and later to look at digit distribution. The question of
the rationality of π was settled in the late 1700s, when Lambert and Legendre proved (using
continued fractions) that the constant is irrational. The question of whether π was algebraic
was settled in 1882, when Lindemann proved that π is transcendental. But this was known
as far back as Aristophanes in 414 BCE, and thus hardly justified further computations.
So what possible motivation lies behind modern computations of π? One motivation is
the raw challenge of harnessing the stupendous power of modern computer systems. Programming such calculations are definitely not trivial, especially on large, distributed memory
computer systems. There have been substantial practical spin-offs. For example, some new
techniques for performing the fast Fourier transform (FFT), heavily used in modern science and engineering computing, had their roots in attempts to accelerate computations of
π. And always the computations help in road-testing computers—often uncovering subtle
hardware and software errors.
Pi in the digital age: With the substantial development of computer technology in the
1950s, π was computed to thousands and then millions of digits. These computations were
greatly facilitated by the discovery soon after of advanced algorithms for the underlying
high-precision arithmetic operations. For example, in 1965 it was found that the (then
newly-discovered) fast Fourier transform (FFT) [5, 7] could be used to perform high-precision
multiplications much more rapidly
√ than conventional schemes.
Such methods (e.g., for ÷, x see [5, 6, 7]) dramatically lowered the time required for
computing π and other constants to high precision. We are now able to compute algebraic
244
Jonathan M. Borwein and Mason S. Macklem
values of algebraic functions essentially as fast as we can multiply, OB (M (N )). In spite of
these advances, into the 1970s all computer evaluations of π still employed classical formulae.
In 1973, Guilloud and Boyer used a formula of Euler for arccot, namely
∞
2
X
(n!) 4n
1
x
=
arctan
,
n+1
2
x
n=0 (2 n + 1)! (x + 1)
to compute a million digits of π. Specifically, they used this formula to express π/4 =
12 arctan (1/18) + 8 arctan (1/57) − 5 arctan (1/239) in the efficient form
∞
∞
2
2
X
X
(n!) 4n
(n!) 4n
1
+
1824
−
20
arctan
,
π = 864
239
(2 n + 1)! 325n+1
(2 n + 1)! 3250n+1
n=0
n=0
where the terms of the second series are just decimal shifts of the first.
Truly new types of infinite series formulae, based on elliptic integral approximations, were
discovered by Srinivasa Ramanujan (1887–1920) around 1910, but were not well known (nor
fully proven) until quite recently when his writings were widely published. They are based
on elliptic functions and are described at length in [2, 5, 7]. One of these series is the
remarkable:
√ ∞
1
2 2 X (4k)! (1103 + 26390k)
=
.
(1)
π
9801
(k!)4 3964k
k=0
Each term of this series produces an additional eight correct digits in the result. When
Gosper used this formula to compute 17 million digits of π in 1985, and it agreed to many
millions of places with the prior estimates, this concluded the first proof of (1) , as described
in [4] ! Actually, Gosper first computed the simple continued fraction for π, hoping to
discover some new things in its expansion, but found none.
At about the same time, David and Gregory Chudnovsky found the following rational
variation of Ramanujan’s formula:
1
π
=
12
∞
X
(−1)k (6k)! (13591409 + 545140134k)
k=0
(3k)! (k!)3 6403203k+3/2
Each term of this series produces an additional 14 correct digits. The Chudnovskys implemented this formula using a clever scheme that enabled them to use the results of an
initial level of precision to extend the calculation to even higher precision. They used this
in several large calculations of π, culminating with a then-record computation of over four
billion decimal digits in 1994.
While the Ramanujan and Chudnovsky series are in practice considerably more efficient
than classical formulae, they share the property that the number of terms needed increases
linearly with the number of digits desired: if you want to compute twice as many digits of π,
you must evaluate twice as many terms of the series. Relatedly, the Ramanujan-type series
!3
∞
2n
X
1
42 n + 5
n
=
.
(2)
n
π
16
16
n=0
allows one to compute the billionth binary digit of 1/π, or the like, without computing the
first half of the series, thus foreshadowing some surprising recent results.
Reduced operational complexity algorithms: In 1976, Eugene Salamin and Richard
Brent independently discovered a reduced complexity algorithm for π. It is based on the
The (digital) life of Pi
245
arithmetic-geometric mean iteration (AGM) and some other ideas due to Gauss and Legendre around 1800, although Gauss, nor many after him, never directly saw the connection
to effectively computing π.
√
Quadratic Algorithm (Salamin-Brent). Set a0 = 1, b0 = 1/ 2 and s0 = 1/2. Calculate
p
ak−1 + bk−1
ak =
(A)
bk = ak−1 bk−1
(G)
(3)
2
2a2
ck = a2k − b2k ,
sk = sk−1 − 2k ck
and compute pk = k .
(4)
sk
Then pk converges quadratically to π. Each iteration of the algorithm doubles the correct
digits. Successive iterations produce 1, 4, 9, 20, 42, 85, 173, 347 and 697 good decimal digits
of π, and takes log N operations for N digits. Twenty-five iterations computes π to over
45 million decimal digit accuracy. A disadvantage is that each of these iterations must be
performed to the precision of the final result.
In 1985, Jonathan and Peter Borwein discovered families of algorithms of this type. For
example, here is a genuinely third-order iteration:
√
Cubic Algorithm: Set a0 = 1/3 and s0 = ( 3 − 1)/2. Iterate
3
rk+1 − 1
2
2
rk+1 =
,
sk+1 =
and
ak+1 = rk+1
ak − 3k (rk+1
− 1).
3
1/3
2
1 + 2(1 − sk )
Then 1/ak converges cubically to π.
√ triples the number of correct digits.
√ Each iteration
Quartic Algorithm: Set a0 = 6 − 4 2 and y0 = 2 − 1. Iterate
yk+1
=
1 − (1 − yk4 )1/4
1 + (1 − yk4 )1/4
and
2
ak+1 = ak (1 + yk+1 )4 − 22k+3 yk+1 (1 + yk+1 + yk+1
).
Then 1/ak converges quartically to π. (Note that only the power of 2 or 3 used in ak depends
on k.) This quartic algorithm, with the Salamin–Brent scheme, was first used by Bailey in
1986 and was used repeatedly by Yasumasa Kanada in Tokyo in computations of π over
the past 15 years or so, culminating in a 200 billion decimal digit computation in 1999 (see
Figure 1). Only 35 years earlier in 1963, Dan Shanks—a very knowledgeable participant—
was confident that computing a billion digits was forever impossible. Today it is easy on a
modest laptop.
Back to the future: In December 2002, Kanada computed π to over 1.24 trillion decimal
digits! His team first computed π in hexadecimal (base 16) to 1,030,700,000,000 places,
using the following two arctangent relations:
1
1
1
1
π = 48 tan−1
+ 128 tan−1
− 20 tan−1
+ 48 tan−1
49
57
239
110443
1
1
1
1
+ 28 tan−1
− 48 tan−1
+ 96 tan−1
.
57
239
682
12943
The first formula was found in 1982 by K. Takano, a high school teacher and song writer.
The second formula was found by F. C. W. Störmer in 1896. Kanada verified the results of
these two computations agreed, and then converted the hex digit sequence to decimal. The
resulting decimal expansion was checked by converting it back to hex. These conversions
are themselves non-trivial, requiring massive computation.
This process is quite different from those of the previous quarter century. One reason
is that reduced operational complexity algorithms require full-scale multiply, divide and
square root operations. These in turn require large-scale FFT operations, which demand
huge amounts of memory, and massive all-to-all communication between nodes of a large
π = 176 tan−1
246
Jonathan M. Borwein and Mason S. Macklem
parallel system. For this latest computation, even the very large system available in Tokyo
did not have sufficient memory and network bandwidth to perform these operations at
reasonable efficiency levels—at least not for trillion-digit computations. Utilizing arctans
again meant using many more arithmetic operations, but no system-scale FFTs, and it can
be implemented using ×, ÷ by smallish integer values—additionally, hex is somewhat more
efficient!
Kanada and his team evaluated these two formulae using a scheme analogous to that
employed by Gosper and by the Chudnovskys in their series computations, in that they
were able to avoid explicitly storing the multiprecision numbers involved. This resulted in
a scheme that is roughly competitive in numerical efficiency with the Salamin-Brent and
Borwein quartic algorithms they had previously used, but with a significantly lower total
memory requirement. Kanada used a 1 Tbyte main memory system, as with the previous
computation, yet got six times as many digits. Hex and decimal evaluations included, it
ran 600 hours on a 64-node Hitachi, with the main segment of the program running at a
sustained rate of nearly 1 Tflop/sec.
The use of arctangents by Kanada and his team marks an unexpected return to these classical formulae, which were used throughout pre-digital calculations of π. Recent summaries
of the history of arctangent calculations of π can be found in [8, 10].
Why Pi? Beyond practical considerations lies the abiding interest in the fundamental
question of the normality (digit randomness) of π. Kanada, for example, has performed
detailed statistical analysis of his results to see if there are any statistical abnormalities
that suggest π is not normal, so far ‘no’. Indeed the first computer computation of π and
e on ENIAC was so motivated by John von Neumann. The digits of π have been studied
more than any other single constant, in part because of the widespread fascination with
and recognition of π. Kanada reports that the 10 decimal digits ending in position one
trillion are 6680122702, while the 10 hexadecimal digits ending in position one trillion are
3F89341CD5.
How to compute the N -th digits of π. One might be forgiven for thinking that essentially everything of interest with regards to π has been dealt with. Even insiders are
sometimes surprised by a new discovery. Prior to 1996, most folks thought if you want to
determine the d-th digit of π, you had to generate the (order of) the entire first d digits.
This is not true, at least for hex (base 16) or binary (base 2) digits of π.
In 1996, Peter Borwein, Plouffe, and Bailey found an algorithm for computing individual hex digits of π. It (1) yields a modest-length hex or binary digit string for π, from
an arbitrary position, using no prior bits; (2) is implementable on any modern computer;
(3) requires no multiple precision software; (4) requires very little memory; and (5) has a
computational cost growing only slightly faster than the digit position. For example, the
millionth hexadecimal digit (four millionth binary digit) of π can be found in four seconds
on a present generation Apple G5 workstation.
This new algorithm is not fundamentally faster than the best known schemes if used
for computing all digits of π up to some position, but its elegance and simplicity are of
considerable interest, and is easy to parallelize. It is based on the following at-the-time new
formula for π:
∞
X
4
2
1
1
1
π=
−
−
−
(5)
16i 8i + 1 8i + 4 8i + 5 8i + 6
i=0
The (digital) life of Pi
247
Name
Year
Correct Digits
Miyoshi and Kanada
1981
2,000,036
Kanada-Yoshino-Tamura
1982
16,777,206
Gosper
1985
17,526,200
Bailey
Jan. 1986
29,360,111
Kanada and Tamura
Sep. 1986
33,554,414
Kanada and Tamura
Oct. 1986
67,108,839
Kanada et. al
Jan. 1987
134,217,700
Kanada and Tamura
Jan. 1988
201,326,551
Chudnovskys
May 1989
480,000,000
Kanada and Tamura
Jul. 1989
536,870,898
Kanada and Tamura
Nov. 1989
1,073,741,799
Chudnovskys
Aug. 1991
2,260,000,000
Chudnovskys
May 1994
4,044,000,000
Kanada and Takahashi
Oct. 1995
6,442,450,938
Kanada and Takahashi
Jul. 1997
51,539,600,000
Kanada and Takahashi
Sep. 1999
206,158,430,000
Kanada-Ushiro-Kuroda
Dec. 2002 1,241,100,000,000
Figure 1. Calculations of π
which was discovered using integer relation methods (see [7]), with a computer program that
ran for several months and then produced the (equivalent) relation:
1 5 1
1
π = 4 F 1, ; , −
+ 2 tan−1
− log 5
4 4 4
2
where F(1, 1/4; 5/4, −1/4) = 0.955933837 . . . is a Gaussian hypergeometric function. Surprisingly, the proof of this result is neither long nor inaccessible, and indeed can fit on a
single overhead slide!
The algorithm in action. In 1997, Fabrice Bellard of INRIA computed 152 binary digits
of π starting at the trillionth position. The computation took 12 days on 20 workstations
working in parallel over the Internet. Bellard’s scheme is based on the following variant of
(5):
∞
∞
X
(−1)k
1 X (−1)k
32
8
1
π = 4
−
+
+
,
4k (2k + 1) 64
1024k 4k + 1 4k + 2 4k + 3
k=0
k=0
which permits hex or binary digits of π to be calculated roughly 43% faster than (5).
In 1998 Colin Percival, then a 17-year-old student at Simon Fraser University, utilized 25
machines to calculate first the five trillionth hexadecimal digit, and then the ten trillionth
hex digit. In September 2000, he found the quadrillionth binary digit is 0, a computation
that required 250 CPU-years, using 1734 machines in 56 countries.
A last comment for this section is that Kanada was able to confirm his 2002 computation
in only 21 hours by computing a 20 hex digit string starting at the trillionth digit, and
comparing this string to the hex string he had initially obtained in over 600 hours. Their
agreement provided enormously strong confirmation.
Changing world views. In retrospect, we may wonder why in antiquity π was not measured to an accuracy in excess of 22/7? Perhaps it reflects not an inability to do so but
248
a very different mind set to a modern experimental—Baconian or Popperian—one. In the
same vein, one reason that Gauss and Ramanujan did not further develop the ideas in their
identities for π is that an iterative algorithm, as opposed to explicit results, was not as
satisfactory for them (especially Ramanujan). Ramanujan much preferred formulae like
3
3
√
log (640320) ≈ π
π ≈ √ log (5280) ,
163
67
correct to 9 and 15 decimal places, both of which rely on deep number theory. Contrastingly,
Ramanujan in his famous 1914 paper Modular Equations and Approximations to Pi [2, p.253]
found
1/4
192
2
9 +
= 3.14159265258 · · ·
22
“empirically, and it has no connection with the preceding theory.” Only the marked digit is
wrong. Indeed, much life remains in this most central of numbers.
Acknowledgement
Thanks are due to many, especially Peter Borwein and David Bailey.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Jorg Arndt and Christoph Haenel, Pi Unleashed (Springer-Verlag New York 2001).
L. Berggren, J.M. Borwein and P.B. Borwein, Pi: a Source Book, third edition (Springer-Verlag 2004).
David Blatner, The Joy of Pi (Walker and Co New York 1997).
J.M. Borwein, P.B. Borwein, and D.A. Bailey, Ramanujan, modular equations and pi or how to compute
a billion digits of pi, MAA Monthly 96 (1989), 201–219. Reprinted in Organic Mathematics Proceedings,
http://www.cecm.sfu.ca/organics, 1996. (Collected in [2].)
J.M. Borwein and P.B. Borwein, Pi and the AGM (John Wiley and Sons 1987).
J.M. Borwein and P.B. Borwein, Ramanujan and Pi, Scientific American, February 1988, 112–117.
Reprinted in pp. 187–199 of Ramanujan: Essays and Surveys, Bruce C. Berndt and Robert A. Rankin
Eds., AMS-LMS History of Mathematics, Vol. 22, 2001. (Collected in [2].)
J.M. Borwein and D.H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century
(AK Peters Ltd 2004).
Hwang Chien-Lih, Some observations on the method of arctangents for the calculation of π, Mathematical Gazette 88 (2004) 270–278.
P. Eymard and J.-P. Lafon, The Number π (American Mathematical Society Providence 2003).
N. Lord, Recent formulae for π: arctan revisited, Mathematical Gazette 83 (1999), 479–483.
There are many Internet resources on Pi, a reliable selection is kept at http://www.
expmath.info.
Department of Computer Science, Dalhousie University, Halifax, Nova Scotia, Canada, B3H1W5.
E-mail: [email protected]
Department of Computer Science, Dalhousie University, Halifax, Nova Scotia, Canada, B3H1W5
E-mail: [email protected]
Received 8 May 2006.